On harmonic functions of symmetric Levy processes

TL;DR

This paper investigates harmonic functions related to certain symmetric Levy processes where standard regularity techniques fail, providing alternative a-priori estimates and analyzing asymptotic behaviors of Green functions and Levy densities.

Contribution

It introduces new a-priori regularity estimates for harmonic functions of Levy processes where classical methods do not apply, and extends existing results on Green functions and Levy densities.

Findings

Established a-priori regularity estimates despite failure of standard methods

Extended asymptotic analysis of Green functions for subordinate Brownian motions

Analyzed Levy densities with slowly varying Laplace exponents

Abstract

We consider some classes of Levy processes for which the estimate of Krylov and Safonov (as in [BL02]) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Holder continuity estimates of harmonic functions. Despite the faliure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from [SSV06] and obtain asymptotic behavior of the Green function and the Levy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.